PREFACE . decided that a certain theorem had really been proved? Might not there be concealed inconsistencies in the very foundations and postulate systems on which the whole elaborate structure had been reared? It began to appear that an exhaustive re- examination of everything from the ground up was demanded* The capital problem was to prove the self-consistency of mathematical analysis — the calculus and its numerous modern offshoots. Presently this programme turned out to be far more difficult than had been anticipated, and David Hilbert (1862- 1943), the last of the giants from the nineteenth century, in 1898 proposed the more modest problem of proving the consis- tency of arithmetic. This led to the like for mathematical logic. All was going well till 1931, when Kurt Godel (1906- ) showed that in any well-defined system of mathematical axioms there exist mathematical questions which cannot be settled on the basis of these axioms. But suppose we go to a more inclusive system in which, perhaps, the questions can be settled. The same difficulty appears in the new system, and so on indefi- nitely. There are thus specific purely mathematical 'yes-no' questions which will be forever undecidable by human beings. This wholly unexpected conclusion has been called the most significant advance in logic since Aristotle. It does not mean that mathematics has gone to smash, but it does suggest that some of the claims made for mathematics in the past will have to be moderated. One philosophical die-hard who thoroughly misunderstood what Godel had done, proudly proclaimed, *I am an Aristotelian. The old logic is good enough for me', which sounded like an echo of the revivalist hymn "The old-time religion, the old-time religion is good enough for me.1 Aristo- telian logic may be good enough for the old-timers, but it is not good enough for mathematics, nor has it been for at least three centuries. As one detail, Aristotle's logic makes no pro- vision for variables and functions as they occur in mathematics* There is not space here to elaborate any of this, but those interested will find an elementary and lucid account by Alfred! • Tarski in his Introduction to Logic and the Methodology of Deduc- tive Sciences (Oxford University Press, 2nd Edition, 1941). 1953 E . T. B E Xi L